Permutative categories, multicategories, and algebraic K-theory

TL;DR

This paper extends the K-theory construction to a broader class of categories, specifically from permutative categories to multicategories, while preserving multiplicative structure in a symmetric monoidal closed bicomplete setting.

Contribution

It introduces a new source category of (based) multicategories for the K-theory construction, generalizing the previous permutative categories, and maintains the multiplicative structure.

Findings

K-theory construction extends to multicategories

Preserves multiplicative structure in the new setting

Fully faithful embedding of permutative categories into multicategories

Abstract

We show that the $K$-theory construction of arXiv:math/0403403, which preserves multiplicative structure, extends to a symmetric monoidal closed bicomplete source category, with the multiplicative structure still preserved. The source category of arXiv:math/0403403, whose objects are permutative categories, maps fully and faithfully to the new source category, whose objects are (based) multicategories.